In a mathematical study with collaborators Tsaneva-Atanosova and Osinga of the University of Bristol, we linked families of models for bursting in pituitary somatotrophs and lactotrophs and in pancreatic beta cells to expose their similarities and differences. Both classes of cells exhibit bursts of action potentials from a depolarized plateau, which is potent at driving calcium entry into the cells for secretion of their respective hormones. We had shown in previous work that in the pituitary models the spikes were transients due to slow attraction to the upper steady state that defines the plateau, whereas in the beta cells the spikes are sustained oscillations that would persist if calcium were fixed rather than slowly varying. The transient character of the spikes in the pituitary case implies that the spikes can only be seen if calcium is in fact changing sufficiently rapidly. This raises an issue for the standard mathematical analysis, which views the activity of the fast variables that control spiking as in quasi-steady state with calcium and hence assumes that calcium changes very slowly. Our solution to the problem was to analyze the behaviors of the full system, including both the fast spiking variables and calcium. We examined the range of behaviors as calcium is made slower and slower, which increases the number of spikes, whether sustained or transient, that can happen during the plateau and characterized the bifurcations (transitions between types of dynamics, including chaos) that occur as each spike is added. Although the results are rather technical, it can perhaps be appreciated that there is a digital-analog issue here, in that it is not obvious how to increase the numbers of spikes discretely by continuously varying a parameter. We showed that at this level there is indeed a strong family resemblance between the pituitary and beta-cell type models as well as clear differences. In this study we employed an abstract and very simple model which means that the results are applicable to other cell types that exhibit plateau bursting or may be found to do so in the future. The simplicity of this model also allowed us to find a generic way to convert one type of burster to the other that is equivalent to shifting the threshold voltage at which the calcium channels activate. This conforms nicely to one's expectation that the models should be closely related since the cell types they model are developmental and evolutionary cousins. This work also points toward a classification scheme that encompasses essentially all the known types of bursting, analogous to the periodic table of elements. We expect to be able to report next year further exciting developments along this line, which largely complete a program of analysis of bursting begun in this lab by former chief John Rinzel 25 years ago. See Ref. # 1. Together with collaborators Bertram (FSU) and Matveev (NJIT) we have written a comprehensive review along with new results on calcium cooperativity of synaptic release (Ref. # 2). Cooperativity is the nonlinear increase in release as external calcium or calcium current is increased. Although others had recognized that these two types of cooperativity are distinct, we developed precise mathematical definitions to help dispel confusion in the literature about what was being measured. Often, what experimenters really want to know is the number of calcium channels that contribute to the exocytosis of a single neurotransmitter vesicle. Indeed, there is a longstanding controversy over whether release is controlled by one or a few channels or whether it is controlled by a large number of channels. Unfortunately, this cannot be measured directly. In our review, we discuss when this cooperativity of calcium channels can be reasonably well approximated by the cooperativity of calcium current and when it cannot. Our survey of the literature shows that both extremes as well as intermediate cases occur in different synapses, and, moreover, that the degree of cooperativity can change with development as a result of changes in the morphology of the presynaptic active zone or of intrinsic biochemical changes in the synaptic machinery. We have extended our kinetic model of the P2X7 receptor, a ligand-gated ion channel activated by extracellular ATP. This receptor is found in pituitary cells and also in macrophages. Brief single stimulations at low agonist concentrations activate small currents and are associated with cell growth and proliferation, whereas repeated stimuli or high agonist concentrations activate much larger currents and lead to cell blebbing and death. Our previous work (see 2010 report) with the Stojilkovic lab (NICHD) showed that a relatively simple Markov model with 8 states was adequate to account for dilation of the ligand-gated channel to a high conductance state after prolonged or repeated exposure to ATP without need for an accessory channel protein, as had been proposed by others. The current work focused on the affect of extracellular calcium to accelerate receptor deactivation when the agonist is removed. The experiments showed that binding of calcium to an allosteric regulatory site was involved and not merely a reduction of free ATP availability in the extracellular space. This hypothesis was supported by the revised mathematical model, in which the forward rates for ATP binding were slowed and the backward rates were accelerated and which could simulate a variety of experimental protocols in detail. See Ref. # 3. In current work we are extending the P2X7 model to P2X2 receptors, with an eye to establishing a unified model for all members of this family.